Safe operation is crucial for deploying robots in human-centered 3D environments. Soft continuum manipulators offer passive safety through mechanical compliance but require active control for reliable collision avoidance. This paper presents a closed-form Control Lyapunov Function and Control Barrier Function controller for real-time 3D obstacle avoidance in soft continuum manipulators without online optimization. The method analytically embeds safety constraints into control inputs, ensuring stability and safety while avoiding feasibility issues of optimization-based approaches.